Method and apparatus for finance-based scheduling of construction projects

ABSTRACT

a method and apparatus is disclosed for scheduling construction projects based on the available finance using integer programming. This method renders CPM/PERT schedules of construction projects executable using bank overdrafts of specified credit limits. Conveniently, the method is organized in three stages; input preparation, schedule extension, and model formulation. Input preparation stage supports preparing CPM/PERT schedule and financial data of the project. Schedule extension stage supports developing a scheme for schedule extension. The scheme is a framework of the original schedule that allows a definite extension increment in the critical path of the CPM. Model formulation stage supports building an integer programming model for the scheme and involves the components of formulating an objective function, setting constraints, and searching for a model solution. The model solution determines the activities&#39; shifts that fulfill the constraints of the specified credit limit while minimizing the schedule extension. A search for a solution of the model is performed. If no solution is found, a repetition of the last two stages of the method is performed after allowing longer extension increment.

BACKGROUND

[0001] 1. Field of Invention

[0002] This invention relates to scheduling systems, specifically tosuch systems which are used for scheduling construction projects.

[0003] 2. Description of the Prior Art

[0004] A crucial factor for construction contractors to run profitablebusiness represents the ability to timely procure adequate fund toexecute construction activities with minimum financing cost. During anyperiod of a project, contractors will never carry out any activity theavailable finance can not afford despite the obligation to adhere to theproject schedule. This explicit principle of operation makes theestablishment of a balance, along project duration, between activities'disbursement and available finance a very vital concept to produce arealistic schedule of project activities.

[0005] The establishment of bank overdrafts has been one of theprevalent methods of financing construction projects. Practically,contractors establish a bank overdraft with a specified credit limit tofinance a new project. Then, they proceed with the execution of projectactivities such that the disbursements during any period, week or month,will never cause negative cash flow to exceed the credit limit of theestablished overdraft. An additional component of financing during aperiod represents a receipt collected as a reimbursement for workperformed during a previous period of the project. Contractors usuallydeposit these components in the overdraft account to avoid that thenegative cash flow reaches the credit limit and also to reduce thefinancing cost. Accordingly, contractors are always interested indeveloping schedules that make project negative cash flow never exceedsthe specified credit limit. Since this concept of scheduling is based onfinance availability, it can be referred to as finance-based scheduling.

[0006] There are many techniques that were developed in the literatureto schedule construction projects based on different concepts. CPM/PERTscheduling techniques were developed with an underlying concept ofminimizing the project duration. Subsequently, many algorithms have beendeveloped to enhance their usefulness. Among these algorithms, resourcemanagement techniques offer a substantial enhancement. The literatureinvolves many heuristic, optimal and sub-optimal methods that have beendeveloped to modify CPM/PERT schedules in account of practicalconsiderations of project resources. Despite the fact that money is amajor construction resource, none of the resource management techniquesconsidered the modification of CPM/PERT schedules to balance the projectexpenditures with the available finance along the project duration.

[0007] Generally, traditional resource management models with theavailable finance treated as a constrained resource can not beconsidered as a substitute for the proposed finance-based schedulingmethod. This is due to the fact that part of the finance availableduring a given period is generated as a reimbursement for a work thatwas performed during a previous period of the same project. Thus, acomplete profile of finance available along the project duration, whichis a required input to these models, can not be readily established.

[0008] On the other hand, resource-oriented scheduling techniquesincluding line of balance (LOB) and linear scheduling method (LSM) areunderlied by the concept of efficient use of resources. Operating underthis concept necessitates the practice of some sort of central controlover resources. As a matter of fact, central control over resourcesmight exist in the military or job shop scheduling in a factory, butdoes not exist in construction projects. Instead, contractors andsubcontractors allocate resources mainly based on the availability offinance. They determine the pace of work considering the efficient useof resources provided that finance availability constraint is fulfilled.

[0009] As outlined before, although many techniques were developed toschedule construction projects, these techniques suffer from a number ofdisadvantages:

[0010] 1- None of the present scheduling techniques includes in itsunderlying concept or algorithm any consideration for the establishmentof a balance between the disbursement of the scheduled activities andthe available finance during a given period. Consequently, the availablescheduling techniques are possibly to produce financially nonrealisticschedules.

[0011] 2- None of the present scheduling techniques shows how to developschedules that correspond to overdrafts of desired credit limits. Lackof control on overdraft credit limits hinders contractors to negotiategood deals in obtaining bank overdrafts.

[0012] 3- None of the present scheduling techniques relates theschedules of all existing and planned projects, In case of a contractorimplementing many projects simultaneously, to the overall liquiditysituation of the contractor.

[0013] 4- None of the present scheduling techniques can help owners askfor a schedule that produces a desired progress payment requirementsscheme that serves his financial interests.

[0014] Objects and Advantages:

[0015] Accordingly, several objects and advantages of my finance-basedproject scheduling system are:

[0016] 1- Establishes financial feasibility in the scheduling process bysimultaneously relating the disbursements of scheduled activities duringperiods to the finance available in the corresponding periods. In otherwords cause the negative cash flow always undervalues the specifiedcredit limit.

[0017] 2- Enables developing schedules that correspond to overdrafts ofdesired credit limits. The control on overdraft credit limit providesmany benefits including:

[0018] Negotiating lower interest rates with banks,

[0019] Setting favorable terms of repayment,

[0020] Reducing penalties incurred for any unused portion of anoverdraft.

[0021] Avoiding the phenomenon of spiraling down. This situation occurswhen the finance available in a given period does not allow schedulingmuch work. During the next period at which a reimbursement is expected,the generated income will allow scheduling less work, and so forth.

[0022] 3- Relates schedules of all existing and planned projects, Incase of a contractor implementing many projects simultaneously, tooverall liquidity situation of the contractor. This brings about manybenefits including:

[0023] insures that projects with large negative cash flows do notcompound each other. Upon contrast, when surplus cash is available, themethod utilizes this cash in scheduling additional work.

[0024] Helps contractors make key decisions regarding bidding for newprojects, determining construction rates, and specifying optimumcommencement dates of new projects.

[0025] 4- Helps owners adjust the profile of financial commitments tocontractors to suit their capabilities. When bids are awarded, ownerscan ask contractors to estimate the progress payment requirements alongthe project duration. Then, owners can ask for a schedule that producesa desired progress payment requirements scheme that serves financialinterests.

[0026] Further objects and advantages are to provide an optimizationtechnique, integer programming, which provides optimum solution as aschedule that fulfills financing constraint of the specified creditlimits and still minimizes the project duration. Still further objectsand advantages will become apparent from a consideration of the drawingsand ensuing description.

DRAWING FIGURES

[0027]FIG. 1 The stages of the scheduling method.

[0028]FIG. 2 A schematic diagram of a typical activity k in a CPMnetwork.

[0029]FIG. 3 A typical cash flow profile for a construction project.

[0030]FIG. 4 The financial terms of a period t.

[0031]FIG. 5 The CPM network with extension increment M, and extendedduration.

[0032]FIG. 6 The logic involved in the scheduling method.

[0033] FIGS. 7A, and 7B are an example CPM schedule and its scheme.

DESCRIPTION OF THE PREFERRED EMBODIMENT

[0034]FIG. 1 illustrates the use of the method to produce finance-basedschedules for construction projects. Even though this is the preferredembodiment, there is nothing about the invention that precludes it frombeing implemented in various manifestations other than the one describedhere.

[0035] In order to support the invention function, the method iscomprised of three primary stages as shown in FIG. 1.

Input Preparation

[0036] The input preparation stage (as seen in stage 10 of FIG. 1)supports preparing CPM/PERT schedule and financial data of the projectgenerally designated by 11 which involves activities 12, sequences ofactivities 13, time data 14, cash out data 15, cash in data 16, and acash flow profile 17. Time data involves durations, early start times,early finish times, and total floats of activities. Cash out datainvolves identification of cost elements and determination of dailydisbursement rates of activities. Cash in data involves prices, timingof receipts, and deduction percentage. The cash flow profile plots cashin and cash out transactions against time. The terminology of theprevious schedule and financial data is described in detail in FIG. 2,FIG. 3, and FIG. 4.

[0037]FIG. 2 represents a schematic diagram of a typical activity k 31in a CPM schedule. Activity k has a duration D_(k) 32, a free floatFF_(k) 33, a total float TF_(k) 34, an early start time ES_(k) 35, anearly finish time EF_(k) 36, and a disbursement rate R_(k) 37 whichincludes direct costs of material, labor, equipment, and subcontractors,and indirect costs of overheads, bonds, and taxes; k=1,2, . . . , nwhere n is the number of project activities (not shown in FIG. 2). Anactivity q 38, designated by a hatched bar in FIG. 2, is arepresentative activity of the set of activities Q_(k) (not shown inFIG. 2) that depend on the early finish time of activity k. Activity qhas a duration D_(q) 39, a free float FF_(q) which is the same as totalfloat TF_(q) and designated by 40, an early start time ES_(q) 41, and anearly finish time EF_(q) 42. An activity n 43, designated by a solid barin FIG. 2, is a representative activity of project critical activities,at the same time it is the terminating activity in the CPM schedule.Activity n has a duration D_(n) 44, an early start time ES_(n) 45, andan early finish time EF_(n) 46. A duration T 47 of CPM schedule is thetotal number of working days, and i 48 denotes to a typical working dayof the schedule.

[0038] Let total disbursements of all activities performed in day i bedenoted by y_(i); this will be referred to as project disbursements atday i. Thus; $\begin{matrix}{y_{i} = {\sum\limits_{p = 1}^{n_{i}}\quad y_{pi}}} & {{Eqn}\quad (1)}\end{matrix}$

[0039] Where; n_(i) is the number of activities that have theirdurations and float values overlapping with day i; and y_(pi) is thedisbursement rate of activities in terms of their R values as isformulated in Equations 16 below.

[0040] A cash flow profile 17 is shown in FIG. 3 for a constructionproject of L 49 periods. The cash flow profile is from the contractors'perspective. The equations are presented conforming to the financialterminology used by Au, T. and Hendrickson, C., Profit Measures forConstruction Projects, Journal of Construction Engineering andManagement, Vol. 112, No. 2, pp. 273-286 (1986). A project disbursementduring a typical project period t 50 is represented by E_(t) 51, thetiming of receiving a receipt from owner is at the end of the sameperiod, and a receipt amount is represented by P_(t) 52; where:$\begin{matrix}{E_{t} = {\sum\limits_{i = 1}^{m}\quad y_{i}}} & {{Eqn}\quad (2)}\end{matrix}$

[0041] Where, m is the number of days comprising period t, as will beindicated in FIG. 4, K is a multiplier to account for a deductionpercentage owners cut of receipts. A cumulative cash flow at the end ofperiod t (for t≧1) is U_(t) 53, where;

U _(t) =V _(t−1) +E _(t)  Eqn(4)

[0042] A net cash flow at end of period t after receiving receipt isV_(t) 54. At the end of previous period (t−1) 55, FIG. 3 shows also acumulative cash flow U_(t−1) 56, a receipt at end of the period P_(t−1)57, and a net cash flow V_(t−1) 58, where;

V _(t−1) =U _(t−1) +P _(t−1)  Eqn(5)

[0043]FIG. 3 shows a net profit G 59 achieved at end of the project, anda maximum negative cash flow R 60 reached in this cash flow profile (notall negative cash flows are shown in FIG. 3). It is to be noted that,the above calculations are based on the premise that the contractor paysthe interest charges due at end of each period, where the total interestcharges O_(t) at end of period t is; $\begin{matrix}{O_{t} = {{rV}_{t - 1} + {r\frac{E_{t}}{2}}}} & {{Eqn}\quad (6)}\end{matrix}$

[0044] Where; r is the borrowing rate per period.

[0045] The first component of O_(t) represents the interest per periodon the net cash flow V_(t−1); the second component approximates intereston E_(t).

[0046] However, if the contractor pays the periodical interest on theaccumulated cash flow at a borrowing rate r, FIG. 4 shows that acumulative cash flow at the end of period t, which encompasses m 61days, including accumulated interest charges is designated by F_(t) 62where,

F _(t) =U _(t)+I_(t)  Eqn(7)

[0047] Where; I_(t) 63 is an accumulated interest charges till the endof period t. $\begin{matrix}{I_{t} = {\sum\limits_{l = 1}^{t}\quad {O_{l}\left( {1 + r} \right)}^{t - l}}} & {{Eqn}\quad (8)}\end{matrix}$

[0048] A net cash flow up till end of period t is N_(t) 64 where;

N _(t)F_(t)+P_(t)  Eqn(9)

[0049] For period t−1, a cumulative cash flow at the end of this periodincluding accumulated interest charges is F_(t−1) 65, and a net cashflow is N_(t−1) 66.

Schedule Extension

[0050] Developing CPM schedules that are constrained with a specifiedcredit limit involves extension rather than compression in theseschedules. Practically, numerous extended schedules could be producedfor a given schedule. Thus, a fundamental objective of the method is tominimize extension in schedules. A schedule extension stage 18 of themethod (as seen in FIG. 1) supports developing a scheme for scheduleextension generally designated by 19. The scheme (as illustrated in FIG.5) is a structure of the original schedule that allows a definiteextension increment M 20 in the critical path of the CPM schedule, addsthe extension increment to original project duration to determine anextended duration (T+M) 21, and extends the total floats of activitiesby the extension increment to produce adjusted total floats J_(k) 22.This can be expressed as:

J _(k) =M+TF _(k)  Eqn(10)

[0051] The adjusted total float is the time space within which anactivity can be shifted without affecting the extended duration of theschedule. For instance, in FIG. 5, activity k can be shifted all the wayto the end of its illustrated adjusted total float. Consequently,activity n should be shifted to the end of its adjusted total float tomaintain the relation that activity n depends on activity k. Thus, theshift of activity k can be done without causing further extension beyond(T+M). The same scenario is valid between activities q and n.

[0052] The scheme represents a defined framework of definite boundarieswhere a solution can be searched. Thus, searching for a solution withina scheme transforms the boundless problem of extending schedules tofulfill finance constraints to a bounded and defined problem that can bemodeled.

Model Formulation

[0053] The model formulation stage 23 (as seen in FIG. 1) supportsbuilding an integer programming model for the scheme generallydesignated by 24 and involves the components of formulating an objectivefunction 25, setting constraints 26, and searching for a model solution27. Constraints involve activity shifting, activity sequence, andspecified credit limit. The model solution determines the activities'shifts that minimizes the total duration fulfilling the constraints of,activity shifting, and activity sequence and specified credit limit.

[0054] The logic of searching for a solution in the present invention isoutlined in FIG. 6 which comprises an application of previouslydiscussed three-stage model, generally designated by 67. A search 68 fora solution of the model is performed. If no solution is found, arepetition of the last two stages of the method is performed, generallydesignated by 69. The repetition process involves, allowing longerextension increment, adjusting total floats, formulating a model for thenew scheme, and searching for a solution. Thus, searching for a solutionat a specified credit limit is an iterative process, each iteration useslarger extension increment than the previous iteration. If manyiterations are performed with no solution that indicates an insufficientcredit limit.

[0055] As described above there are seven unique capabilities in theinvention:

[0056] 1. Produces realistic schedules as far as finance availability isconcerned and thus increases the utility of schedules.

[0057] 2. Builds on the widely-spread CPM and PERT techniques and thusare readily plausible for practitioners.

[0058] 3. Achieves financial feasibility besides the demanded goal oftime minimization.

[0059] 4. Employs the optimization technique of integer programming togive optimum solutions.

[0060] 5. Defines a scheme for the extension of schedules to achieveminimum extension of the schedule total duration.

[0061] 6. Defines a scheme that transforms the process of seeking anextended schedule that fulfils finance constraints from a searching in aboundless region to a search in a well-defined and definite region.

[0062] 7. Provides a sensitivity analysis technique through the gradualincrease in extension increment.

[0063] These are not dependent on each other. The integer programming,for instance, could be used with scheduling techniques other than CPM orPERT, without defining a scheme, or without incremental increase of thesearching scope, perhaps not so efficiently. In this embodiment, theseven major features of the invention have been united to form anefficient and compact combination as described herein.

[0064] The following sections illustrate the manner in which theinvention has been implemented in the preferred embodiment. Theimplementation of the method involves basically the model formulation.Thus, the model formulation is explained with an illustrative example.

[0065] The model is formulated by considering the previously introducedvariables and the general activity k. The proposed model employedCPM/PERT constraints of activity sequence and activity shifting asdeveloped by Easa, S. M., Resource Leveling in Construction ByOptimization, Journal of Construction Engineering and Management, Vol.115, No. 2, pp. 302-316., (1989). Moreover, the present invention addsan additional constraint to cover the financial aspect.

[0066] Objective Function

[0067] The decision variable x denotes the shift in an activity. Asmentioned above, the objective of the model is to minimize the totalextension of the schedule through minimizing the shifting in the lastcritical activity n. Thus, the objective function can be formulated as:

Minimize z=x_(n)  Eqn(11)

[0068] Activity shifting constraints

[0069] This insures that the shifting in activities x_(k); k=1, 2, . . ., n where n is the total number of activities, is an integer within theadjusted total float. The amount of shifting of activities x_(k) can berepresented by: $\begin{matrix}{{x_{k} = {{\sum\limits_{j = 1}^{J_{k}}\quad {j\quad S_{kj}\quad k}} = 1}},2,\quad \ldots \quad,n} & {{Eqn}\quad (12)} \\{{{{\sum\limits_{j = 1}^{J_{k}}\quad S_{k\quad j}}\quad \leq {1\quad k}} = 1},2,\quad \ldots \quad,n} & {{Eqn}\quad (13)}\end{matrix}$

[0070] where J_(k)=adjusted total float of activity k. S_(kj)ε{0, 1} arebinary variables. These equations ensure that x_(k) takes values of 0,1, 2, . . . , J_(k) and only one of these values is considered at atime. In other words, Eqn. 13 means that only one S_(kj) equals I andthe others equal zero (in this case x_(k)=j_(k)) or all S_(kj) equalzero (in this case x_(k)=0). Thus, Eqns. 12 and 13 allow x_(k) to takeall possible values from 0 up to J_(k).

[0071] Activity Sequence Constraints

[0072] A constraint between activity k and each of the activitiesqεQ_(k) is needed. This is because the early finish time of eachactivity q must be equal to or more than the earliest finish of activityk plus the duration of activity q. This can be representedmathematically by:

EF _(q)≧(EF _(k) +D _(q)) k=1, 2, . . . , n for all qεQ_(k)  Eqn(14)

[0073] Specified Credit Limit Constraints

[0074] The negative cash flow at any period t including accumulatedinterest charges F_(t) should not exceed the specified credit limit W ofthe overdraft. This constraint can be formulated as follows:

F_(t)≦W  Eqn(15)

[0075] F_(t), as in Equation 7, for each period of the project can beformulated in terms of disbursement rate y_(pi) using equations 1 till 8respectively. The disbursement rate y_(pi) can be formulated similar tothe formulation of resource rate used by Easa, S. M., Resource Levelingin Construction By Optimization, Journal of Construction Engineering andManagement, Vol. 115, No. 2, pp. 302-316., (1989) as in equations 16.Note that Equations 16 uses the subscript k to illustrate activity k,but it should be used for all activities p which are performed in timeunit i.

[0076] For activity k of disbursement rate R_(k), the disbursement ratein time unit i can be calculated as follows: $\begin{matrix}{{y_{ki} = {\left( {1 - {\sum\limits_{j = 1}^{J_{k}}S_{k\quad j}}} \right)\quad R_{k}}};{{E\quad S_{k}}; \leq i < {E\quad F_{k}}}} & {{Eqn}\quad \left( {16a} \right)}\end{matrix}$

y _(ki) =S _(kj) .R _(k) ; ES _(k) +j≦i<EF _(k) +j j=1,2, . . . ,J_(k)  Eqn(16b)

y_(ki)=0; otherwise  Eqn(16c)

[0077] As noted from Eqn. 16a, when ΣS_(kj)=0 (i.e., x_(k)=0 from Eq.12) the disbursement rate R_(k) will be considered for each time unitwithin ES_(k) and EF_(k). As also noted from Eq. 16b, when any S_(kj)=1(i.e., x_(k)=j_(k) from Eqns. 12 and 13) the disbursement rate R_(k)will be considered for each time unit in which activity k is performedin its shifted position.

[0078] Finally, the objective function and constraints are formulated interms of S_(kj) variables. Thus, the model constraints and objectivefunction represents an integer optimization model that can be solveddirectly by a number of the existing computer software systems. The nonzero variable indicates the shift, for instance, if Sk_(k3)=1 thatindicates a shift value in activity k of 3 time units.

EXAMPLE

[0079] An illustrative CPM schedule example is used to demonstrate themethod presented in this invention. FIG. 7A shows the original scheduleconsisting of five activities, disbursement rate ($/day) is indicatedabove each activity. A scheme is shown in FIG. 7B with 5-day extensionincrement. The extended duration is 16 working days which representsthree full periods each comprises five working days and a fourth periodof one day. The objective function and constraints are presented below:

[0080] Objective Function:

[0081] Using Equation 12 to formulate the shift in the last activity Eof the schedule and substituting in Equation 11, the objective functionof the example schedule becomes:

Minimize Z=S _(E1)+2S _(E2)+3S _(E3)+4S _(E4)+5S _(E5)  Eqn(17)

[0082] Activity Sequence Constraints:

[0083] Referring to FIG. 7A, equation 14 can be interpreted in terms ofshifts as follows for activities A, B, C, D, and E.

[0084] Shift of activity A should not be greater than shift of activityE by three days which is the total float of A.

[0085] Shift of activity B should not be greater than shift of activityC.

[0086] Shift of activity B should not be greater than shift of activityD.

[0087] Shift of activity C should not be greater than shift of activityE.

[0088] Shift of activity D should not be greater than shift of activityE by three days which is the total float of D. This is to insure thatthe schedule duration remains determined by end of activity E.

[0089] Using equation 12 to formulate shifts of activities A, B, C, D,and E, the above five constraints can respectively be formulated asfollows: $\begin{matrix}\begin{matrix}{S_{A1} + {2S_{A2}} + {3S_{A3}} + {4S_{A4}} + {5S_{A5}} + {6S_{A6}} + {7S_{A7}} + {8S_{A8}} -} \\{{S_{E1} - {2S_{E2}} - {3S_{E3}} - {4S_{E4}} - {5S_{E5}}} \leq 3}\end{matrix} & {{Eqn}\quad (18)} \\{{S_{B1} + {2S_{B3}} + {3S_{B3}} + {4S_{B4}} + {5S_{B5}} - S_{C1} - {2S_{C2}} - {3S_{C3}} - {4S_{C4}} - {5S_{C5}}} \leq 0} & {{Eqn}\quad (19)} \\\begin{matrix}{S_{B1} + {2S_{B3}} + {3S_{B3}} + {4S_{B4}} + {5S_{B5}} - S_{D1} - {2S_{D2}} - {3S_{D3}} - {4S_{D4}} -} \\{{{5S_{D5}} - {6S_{C6}} - {7S_{D7}} - {8S_{D8}}} \leq 0}\end{matrix} & {{Eqn}\quad (20)} \\{{S_{C1} + {2S_{C2}} + {3S_{C3}} + {4S_{C4}} + {5S_{C5}} - S_{E1} - {2S_{E2}} - {3S_{E3}} - {4S_{E4}} - {5S_{E5}}} \leq 0} & {{Eqn}\quad (21)} \\\begin{matrix}{S_{D1} + {2S_{D2}} + {3S_{D3}} + {4S_{D4}} + {5S_{D5}} + {6S_{D6}} + {7S_{D7}} + {8S_{D8}} - S_{E1} -} \\{{{2S_{E2}} - {3S_{E3}} - {4S_{E4}} - {5S_{E5}}} \leq 3}\end{matrix} & {{Eqn}\quad (22)}\end{matrix}$

[0090] Activity Shifting Constraints:

[0091] Referring to equation 13, activity shifting constraints can beformulated as follows for activities A, B, C, D, and E.

S _(A1) +S _(A2) +S _(A3) +S _(A4) +S _(A5) +S _(A6) +S _(A7) +S_(A8)≦1  Eqn(23)

S _(B1) +S _(B3) +S _(B3) +S _(B4) +S _(B5)≦1  Eqn(24)

S _(C1) +S _(C2) +S _(C3) +S _(C4) +S _(C5)≦1  Eqn(25)

S _(D1) +S _(D2) +S _(D3) +S _(D4) +S _(D5) +S _(D6) +S _(D7) +S_(D8)≦1  Eqn(26)

S _(E1) +S _(E2) +S _(E3) +S _(E4) +S _(E5)≦1  Eqn(27)

[0092] Specified Credit Limit Constraints:

[0093] Referring to FIG. 7B, the project daily disbursement y_(i) inequation 1 can be formulated for 16 days using equations 16a, 16b, and16c as follows: $\begin{matrix}\begin{matrix}{y_{1} = {5000 - {2000S_{A1}} - {2000S_{A2}} - {2000S_{A3}} - {2000S_{A4}} - {2000S_{A5}} - {2000S_{A6}} -}} \\{{2000S_{A7}} - {2000S_{A8}} - {3000S_{B1}} - {3000S_{B2}} - {3000S_{B3}} - {3000S_{B4}} - {3000S_{B5}}}\end{matrix} & {{Eqn}\quad (28)} \\\begin{matrix}{y_{2} = {5000 - {2000S_{A2}} - {2000S_{A3}} - {2000S_{A4}} - {2000S_{A5}} - {2000S_{A6}} - {2000S_{A7}} -}} \\{{2000S_{A8}} - {3000S_{B2}} - {3000S_{B3}} - {3000S_{B4}} - {3000S_{B5}}}\end{matrix} & {{Eqn}\quad (29)} \\\begin{matrix}{y_{3} = {5000 - {2000S_{A3}} - {2000S_{A4}} - {2000S_{A5}} - {2000S_{A6}} - {2000S_{A7}} - {2000S_{A8}} -}} \\{{3000S_{B3}} - {3000S_{B4}} - {3000S_{B5}}}\end{matrix} & {{Eqn}\quad (30)} \\\begin{matrix}{y_{4} = {8000 - {2000S_{A4}} - {2000S_{A5}} - {2000S_{A6}} - {2000S_{A7}} - {2000S_{A8}} + {3000S_{B1}} + {3000S_{B2}} +}} \\\begin{matrix}{{3000S_{B3}} - {4000S_{C1}} - {4000S_{C2}} - {4000S_{C3}} - {4000S_{C4}} - {4000S_{C5}} - {2000S_{D1}} -} \\{{2000S_{D2}} - {2000S_{D3}} - {2000S_{D4}} - {2000S_{D5}} - {2000S_{D6}} - {2000S_{D7}} - {2000S_{D8}}}\end{matrix}\end{matrix} & {{Eqn}\quad (31)} \\\begin{matrix}{y_{5} = {8000 - {2000S_{A5}} - {2000S_{A6}} - {2000S_{A7}} - {2000S_{A8}} + {3000S_{B2}} + {3000S_{B3}} +}} \\\begin{matrix}{{3000S_{B4}} - {4000S_{C2}} - {4000S_{C3}} - {4000S_{C4}} - {4000S_{C5}} - {2000S_{D2}} -} \\{{2000S_{D3}} - {2000S_{D4}} - {2000S_{D5}} - {2000S_{D6}} - {2000S_{D7}} - {2000S_{D8}}}\end{matrix}\end{matrix} & {{Eqn}\quad (32)} \\\begin{matrix}{y_{6} = {8000 - {2000S_{A6}} - {2000S_{A7}} - {2000S_{A8}} + {3000S_{B3}} + {3000S_{B4}} + {3000S_{B5}} - {4000S_{C3}} -}} \\{{4000S_{C4}} - {4000S_{C5}} - {2000S_{D3}} - {2000S_{D4}} - {2000S_{D5}} - {2000S_{D6}} - {2000S_{D7}} - {2000S_{D8}}}\end{matrix} & {{Eqn}\quad (33)} \\\begin{matrix}{y_{7} = {6000 + {2000S_{A1}} + {2000S_{A2}} + {2000S_{A3}} + {2000S_{A4}} + {2000S_{A5}} + {2000S_{A6}} + {3000S_{B4}} + {3000S_{B5}} -}} \\{{4000S_{C4}} - {4000S_{C5}} - {2000S_{D4}} - {2000S_{D5}} - {2000S_{D6}} - {2000S_{D7}} - {2000S_{D8}}}\end{matrix} & {{Eqn}\quad (34)} \\\begin{matrix}{y_{8} = {6000 + {2000S_{A2}} + {2000S_{A3}} + {2000S_{A4}} + {2000S_{A5}} + {2000S_{A6}} + {2000S_{A7}} +}} \\{{3000S_{B5}} - {4000S_{C5}} - {2000S_{D5}} - {2000S_{D6}} - {2000S_{D7}} - {2000S_{D8}}}\end{matrix} & {{Eqn}\quad (35)} \\\begin{matrix}{y_{9} = {4000 + {2000S_{A3}} + {2000S_{A4}} + {2000S_{A5}} + {2000S_{A6}} + {2000S_{A7}} + {2000S_{A8}} +}} \\{{2000S_{D1}} + {2000S_{D2}} + {2000S_{D3}} + {2000S_{D4}} + {2000S_{D5}}}\end{matrix} & {{Eqn}\quad (36)} \\\begin{matrix}{y_{10} = {3000 + {2000S_{A4}} + {2000S_{A5}} + {2000S_{A6}} + {2000S_{A7}} + {2000S_{A8}} + {4000S_{C1}} +}} \\{{4000S_{C2}} + {4000S_{C3}} + {4000S_{C4}} + {4000S_{C5}} + {2000S_{D2}} + {2000S_{D3}} + {2000S_{D4}} + {2000S_{D5}} +} \\{{2000S_{D6}} - {3000S_{E1}} - {3000S_{E2}} - {3000S_{E3}} - {3000S_{E4}} - {3000S_{E5}}}\end{matrix} & {{Eqn}\quad (37)} \\\begin{matrix}{y_{11} = {3000 + {2000S_{A5}} + {2000S_{A6}} + {2000S_{A7}} + {2000S_{A8}} +}} \\{{4000S_{C2}} + {4000S_{C3}} + {4000S_{C4}} + {4000S_{C5}} + {2000S_{D3}} + {2000S_{D4}} + {2000S_{D5}} +} \\{{2000S_{D6}} + {2000S_{D7}} - {3000S_{E2}} - {3000S_{E3}} - {3000S_{E4}} - {3000S_{E5}}}\end{matrix} & {{Eqn}\quad (38)} \\{y_{12} = {{2000S_{A6}} + {2000S_{A7}} + {2000S_{A8}} + {4000S_{C3}} + {4000S_{C4}} + {4000S_{C5}} +}} & {{Eqn}\quad (39)} \\{{2000S_{D4}} + {2000S_{D5}} + {2000S_{D6}} + {2000S_{D7}} + {2000S_{D8}} + {3000S_{E1}} + {3000S_{E2}}} & \quad \\\begin{matrix}{y_{13} = {{2000S_{A7}} + {2000S_{A8}} + {4000S_{C4}} + {4000S_{C5}} + {2000S_{D5}} +}} \\{{2000S_{D6}} + {2000S_{D7}} + {2000S_{D8}} + {3000S_{E2}} + {3000S_{E3}}}\end{matrix} & {{Eqn}\quad (40)}\end{matrix}$

y ₁₄=2000 S _(A8)+4000 S _(C5)+2000 S _(D6)+2000 S _(D7)+2000 S_(D8)+3000 S _(E3)+3000 S _(E4)  Eqn(41)

y ₁₅=2000 S _(D7)+2000 S _(D8)+3000 S _(E4)+3000 S _(E5)  Eqn(42)

y ₁₆=2000 S _(D8)+3000 S _(E5)  Eqn(43)

[0094] Then, equation 2 can be used to formulate Et for the four periodsof the schedule as follows:

E ₁ =y ₁ +y ₂ +y ₃ +y ₄ +y ₅  Eqn(44)

E ₂ =y ₆ +y ₇ +y ₈ +y ₉ +y ₁₀  Eqn(45)

E ₃ =y ₁₁ +y ₁₂ +y ₁₃ +y ₁₄ +y ₁₅  Eqn(46)

E ₄ =y ₁₆  Eqn(47)

[0095] Note that four equations 44 till 47 are in terms of S_(kj). Giventhe value of the multiplier K, the corresponding formulas of P_(t) canbe performed using equation 3. Then, formulas for U_(t), V_(t), I_(t),and F_(t) can be established in terms of S_(kj) using equations 4, 5, 8,and 7 respectively. Accordingly, five constraints can be established toexpress the conditions that F₁ till F₅ must undervalue the specifiedcredit limit. These constraints are not presented here for being toolengthy. It is to be noted that credit limit could be constantthroughout periods or varies from period to period.

[0096] Then, the objective function in equation 17, activity sequenceconstraints in equations 18 till 22, activity shifting constraints inequations 23 till 27, and credit limit constraints represent the integerprogramming model that can be solved for S_(kj) values using availablesoftware.

[0097] Summary, Ramifications, and Scope

[0098] Thus the reader will see that the finance-based scheduling methodof the invention provides a reliable method that can be used to producerealistic schedules as far as finance availability is concerned with thefollowing advantages:

[0099] 1. Increases the usefulness of schedules by rendering themfeasible as far as finance availability is concerned.

[0100] 2. Builds trust on schedules from wider sector of practitionersand thus increases the willingness to use CPM as a project managementtool.

[0101] 3. Offers twofold benefits of minimizing of total projectduration and fulfilling finance availability constraints.

[0102] 4. Offers optimum solutions through employing integer programmingoptimization technique.

[0103] 5. Provides an invaluable tool for negotiating and establishinggood bank overdrafts.

[0104] 6. Introduces CPM technique to the attention of decision makersas a business-management tool as well as a production-management tool.

[0105] 7. Brings businessmen and production men to common grounds ofunderstanding so that their goals are unified rather than conflicting.

[0106] Although the description above contains many specificities, theseshould not be construed as limiting the scope of the invention but asmerely providing illustrations of some of the presently preferredembodiments of this invention. For example, the optimization techniquecan be replaced by a suboptimal, heuristic technique or any algorithmthat fulfills the constraint of specified finance; The method can handlecredit limits that vary in values along schedule periods, thus caneasily handle additional short-term loans; the method can achieve otherbenefits rather than time minimization; the concept of finance-basedscheduling can be applied to the already existing scheduling methods orany new techniques that will be developed in the future based ondifferent concepts of scheduling, the concept of finance-basedscheduling can be used for projects rather than construction projectswhich have similar features as far as financing is concerned, etc.

[0107] Thus the scope of the invention should be determined by theappended claims and their legal equivalents, rather than by theembodiment illustrated.

I claim: 1- A method for revising a CPMP/PERT schedule of a constructionproject that determines revised start times of activities and a revisedduration for completing said project, such that said project can beadequately financed throughout said revised duration using a bankoverdraft of a predetermined credit limit, using integer programmingtechnique, whereby said schedule is rendered executable with respect tothe available finance of said predetermined credit limit comprising thesteps of: devising a scheme that allows an extension of said schedule,formulating a model for said scheme using said integer programmingtechnique, and providing a processor to solve said model and output anoptimum solution. 2- The method of claim I wherein the step of devisingsaid scheme involves adding an extension increment to the duration ofsaid schedule to produce an extended duration for said schedule. 3- Themethod of claim 2 wherein said extension increment is a plurality ofdays that are added to the total float of each of said activities toproduce an adjusted total float. 4- The method of claim 3 wherein saidadjusted total float allows for a shift in each of said activities suchthat negative cash flows at the end of periods of said projectundervalue said predetermined credit limit. 5- The method of claim 4wherein if said shift in each of said activities is within saidrespective adjusted total float, this causes said revised duration toremain below said extended duration. 6- The method of claim I whereinthe step of formulating said model for said scheme using said integerprogramming technique involves setting an objective function and aplurality of constraints. 7- The method of claim 6 wherein saidobjective function of said model sets a goal of minimizing said revisedduration of said project. 8- The method of claim 6 wherein saidplurality of constraints involves constraints of activity shifting,activity sequence, and said predetermined credit limit. 9- The method ofclaim 8 wherein said constraints of activity shifting involve modelingof said shift in each of said activities to take up an integer valuewithin said adjusted total float. 10- The method of claim 8 wherein saidconstraints of activity sequence establish and maintain the dependencybetween said activities. 11- The method of claim 8 wherein saidconstraints of said credit limit insure that said negative cash flows atthe end of said periods undervalue said predetermined credit limit. 12-The method of claim 11 wherein said predetermined credit limit could beconstant of different for said periods of said project. 13- The methodof claim 11 wherein said negative cash flows at the end of said periodsinclude accumulated net cash flows at start of said periods plusdisbursements during said periods. 14- The method of claim 12 whereinsaid disbursements involve direct costs of material, labor, equipment,and subcontractors; and indirect costs of overheads, bonds, taxes, andfinancing costs. 15- The method of claim 1 wherein the step of providingsaid processor includes: setting said constraints of said model tointercept a plurality of points each one represents a solution of saidmodel, and applying a searching technique to search for said optimumsolution that minimizes the value of said objective function. 16- Themethod of claim 1 wherein the result of said optimum solution involvesthe value of said shift in each of said activities. 17- The method ofclaim 1, further comprising if no said optimum solution is found,another said scheme is produced considering larger said extensionincrement until said optimum solution is found. 18- An apparatus forrevising a CPM/PERT schedule of a construction project that determinesrevised start times of activities and a revised duration for completingsaid project, such that said project can be adequately financedthroughout said revised duration using a bank overdraft of apredetermined credit limit, using integer programming technique, wherebysaid schedule is rendered executable with respect to the availablefinance of said predetermined credit limit, said computer comprising: amemory to store the data of said schedule including start times, finishtimes, total floats, activity sequences, disbursement rates, andfinancial contract terms, a work area to devise a scheme that allows anextension of said schedule by adding an extension increment of aplurality of days to said total floats of said activities to produceadjusted total floats that allow for shifts in said activities such thatnegative cash flows at the end of periods of said project undervaluesaid predetermined credit limit, an encoded microprocessor to formulatea model for said scheme using said integer programming technique with anobjective function that minimizes said revised duration and a pluralityof constraints involves activity shifting that model said shifts to takeup integer values, activity sequence that establish the dependencybetween said activities, and said predetermined credit limit that insuresaid negative cash flows at the end of said periods undervalue saidpredetermined credit limit, a means for searching an optimum solution ofsaid model that determines values of said shifts and minimizes the valueof said objective function, and a control mechanism that, if no saidoptimum solution is found, devises another said scheme consideringlarger value of said extension increment, formulates said model,searches for an optimum solution, and repeats until said optimumsolution is found. 19- A computer for revising a CPM/PERT schedule of aconstruction project that determines revised start times of activitiesand a revised duration for completing said project, such that saidproject can be adequately financed throughout said revised durationusing a bank overdraft of a predetermined credit limit, whereby saidschedule is rendered executable with respect to the available fund ofsaid predetermined credit limit, said computer comprising: a memory thatstores the data of said schedule including start times, finish times,total floats, activity sequences, disbursement rates, and financialcontract terms, and, microprocessor being encoded with programminginstructions, said instructions adapted to execute minimum extension ofsaid schedule through shifting said activities to fulfill the constraintthat said schedule is executable with said predetermined credit limit.